Results 1 to 3 of 3

Thread: Lebesgue measure on sigma-fields of Borel sets

  1. #1
    Member
    Joined
    Mar 2008
    From
    Acolman, Mexico
    Posts
    118

    Lebesgue measure on sigma-fields of Borel sets

    Hello, I am completely lost with this problem. Any guidance is greatly appreciated.

    Let $\displaystyle \Omega=\{(x,y):0<x\leq 1,0<y\leq 1\}$, let $\displaystyle \mathcal{B}^2$ be the $\displaystyle \sigma$-field of Borel subsets of $\displaystyle \Omega$ and let $\displaystyle P=\lambda_2$ be the Lebesgue measure on $\displaystyle \mathcal{B}^2$. Let $\displaystyle A_0=\{(x,y):x>0,y>0,x+y\leq 1\}$.

    a. Starting from the fact that $\displaystyle \mathcal{B}^2=\sigma(\mathcal{R}^{0,2})$, show that $\displaystyle A_0 \in \mathcal{B}^2$, where $\displaystyle \mathcal{R}^{0,2}$ is the collection of two-dimensional bounded rectangles of the form$\displaystyle B(a_1,b_1,a_2,b_2) = \{(x,y):a_1<x \leq b_1, a_2< y \leq b_2\}$ with $\displaystyle a_1,b_1,a_2,b_2 \in [0,1]$, and $\displaystyle a_1 \leq b_1, a_2 \leq b_2$


    b. Let $\displaystyle P_*$ and $\displaystyle P^*$ be the inner and outer measures defined from$\displaystyle P=\lambda_2$. Prove that $\displaystyle P_*(A_0)=P^*(A_0)=P(A_0)=1/2$. (Hint: one way to do this is to construct sequences of sets $\displaystyle B_n$ and $\displaystyle C_n$ of known measure , e.g. unions of disjoint rectangles, with $\displaystyle B_n \downarrow A_0$ and $\displaystyle C_n \uparrow A_0$)

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22

    Re: Lebesgue measure on sigma-fields of Borel sets

    Quote Originally Posted by akolman View Post
    Hello, I am completely lost with this problem. Any guidance is greatly appreciated.

    Let $\displaystyle \Omega=\{(x,y):0<x\leq 1,0<y\leq 1\}$, let $\displaystyle \mathcal{B}^2$ be the $\displaystyle \sigma$-field of Borel subsets of $\displaystyle \Omega$ and let $\displaystyle P=\lambda_2$ be the Lebesgue measure on $\displaystyle \mathcal{B}^2$. Let $\displaystyle A_0=\{(x,y):x>0,y>0,x+y\leq 1\}$.

    a. Starting from the fact that $\displaystyle \mathcal{B}^2=\sigma(\mathcal{R}^{0,2})$, show that $\displaystyle A_0 \in \mathcal{B}^2$, where $\displaystyle \mathcal{R}^{0,2}$ is the collection of two-dimensional bounded rectangles of the form$\displaystyle B(a_1,b_1,a_2,b_2) = \{(x,y):a_1<x \leq b_1, a_2< y \leq b_2\}$ with $\displaystyle a_1,b_1,a_2,b_2 \in [0,1]$, and $\displaystyle a_1 \leq b_1, a_2 \leq b_2$
    Each of those rectangles is borel....soo.

    b. Let $\displaystyle P_*$ and $\displaystyle P^*$ be the inner and outer measures defined from$\displaystyle P=\lambda_2$. Prove that $\displaystyle P_*(A_0)=P^*(A_0)=P(A_0)=1/2$. (Hint: one way to do this is to construct sequences of sets $\displaystyle B_n$ and $\displaystyle C_n$ of known measure , e.g. unions of disjoint rectangles, with $\displaystyle B_n \downarrow A_0$ and $\displaystyle C_n \uparrow A_0$)

    Thanks in advance
    I mean, do you have any idea?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2008
    From
    Acolman, Mexico
    Posts
    118

    Re: Lebesgue measure on sigma-fields of Borel sets

    I have an idea, but there are some gaps I'm not really sure how to fill in.

    Let $\displaystyle B_n = \cup_{i=1}^N\{(x,y):\frac{i-1}{2^n} < x \leq \frac{i}{2^n}, 0 < y \leq 1- \frac{i-1}{2^n}\}$
    By construction$\displaystyle B_{n+1} \subset B_n$ and$\displaystyle A_0 \subset \cap_{n=1}^\infty B_n.$
    I am pretty sure that $\displaystyle \cap_{n=1}^\infty B_n \subset A_0$ but I don't know how to prove it.

    For the second part, if the above is true $\displaystyle B_n \downarrow A_0$ so $\displaystyle P(B_n) \downarrow P(A_0)$ which should be 1/2. since $\displaystyle P=\lambda_2 $and $\displaystyle \mathcal{B}^2$ is a $\displaystyle \sigma$ field the outer and inner measure should agree.

    Is that on the right track? any hints to fill in the gaps? thanks again.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Borel sigma Algebra
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 28th 2011, 08:39 AM
  2. borel sigma algebra
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 17th 2010, 06:14 AM
  3. Borel Sigma Algebras
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Oct 4th 2009, 01:24 PM
  4. Borel-Sigma Algebras
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: Sep 13th 2009, 07:30 AM
  5. Probability measure on Borel-sigma
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: Aug 11th 2009, 07:55 AM

Search Tags


/mathhelpforum @mathhelpforum